PENNSYLVANIA, 
v. L. L. KUHN, PRINTER TO THE COMMONWEALTH 


WH Avie” 
YEARS VII-XII 


HABRISBURG, 


COMMONWEALTH OF PENNSYLVANIA 
DEPARTMENT OF PUBLIC INSTRUCTION 


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COMMONWEALTH OF PENNSYLVANIA 
DEPARTMENT OF PUSLIC INSTRUCTION 
comics 
STATE COUNCIL OF EDUCATION 
President and Chief Executive Officer 
Thomas E. Finegan 
Term Expires 
(Marion: haweards | ParkiPn ay eto vote ria es att eleameters Bryn Mawr July 1928 
Mrs. Edward W. Biddle, BALLARD SAS EARL a ah ci oh an .--Carlisle July 1928 
Homer: De VV STS ore Se ee re an ine an eMart Oe neg Pittsburgh July 1927 
John C., Saye EA aR Bea) ) La rau eae OS paw iotahs Sept hua Philadelphia July 1926 © 
Mire! cB) Si. Ee Me Cantley eens ous beech UtOR e mere Beaver July 1926 
Mere) TS. Rrentisss Nichols aes OU ee Or Re ve ia Philadelphia July 1925 
HM roderiks RASMUSSEN Pees. oni ENING bonds wie eae ee eiaere Harrisburg July 1924 
Fons) 2 MEATCUS  WARPOT ae oes vk ti SGiede al POON Woaia! atediel oe Pittsburgh July 1924 


rnests Laplace, NEI Dale) yee le a as eit ace ek ve Philadelphia July 1923 


Thomas E. Finegan, M.A., Pa:D., Litt.D., L.B.D., LU.D. 


Superintendent of Public Instruction 
J. George Becht, M.A., SeD.. LL.D. ‘ 


James N. Rule, B.S., M.S. 
Albert W. Johnson, B.A, 
Helen J. Ostrander 


Deputy Superintendent, Higher Education 
Deputy Superintendent, Secondary Education 
Assistant to Superintendent (School Law) 


Secretary to Superintendent — 


DIRECTORS OF BUREAUS 
PV HASETA ELON coi U vase Sica Made le wate Wintwia: aa A ; .-Francis B. Haas, B.S., M.A. 
PATTON ANCE, oi nak ones glee Qiedared erie etna SNE cher Ade alk AS W. M. Denison, B.A., M.A. 
FReAIEH PAU CRETOR ooh wis a ois) chute alae bd ste hie tere erate bre Charles H. Keene, B.A., M.D. 
Pre-Professional and Professional Credentials, C. D. Koch, M.A., Pd.D., Litt.D. 
RHPA ROIOAETON, Me sash pune a Matet do ate es ws ol ubaiiila ee eLV aM waiod, Candee, Lee L. Driver, M.A., Li.D. 
School Buildings, FAT RRR cs ne Er) Am MA II HuBert ©. Hicher, Se.B.. A.B., M.S. 
Schoo] Employes’ Rativeiienh Ue SATORU it Pee LRA ay a tok mr 1 o Bged S Oe 'Baish, M.A. 
MDECIAL: MOUCATIOR o's) Gare cient oie aw SNAPS A plaee eeS Ok Dieta line ys Francis N. Maxfield, Ph.D. 
PORCH OT Wicca! wiawue 4 gee ele Wea an alk oie! oe ute: Wlpeig Albert L. Rowland, M.A., Ph.D. 
MOCATIOM AL ee EO a igtie dere, area a WNataibieKiata tT ACL IPRS. a aang L. H. Dennis, B.S. 


DIRECTORS OF SUBJECTS 


Apt FUCUCRTLOM, 0 OS ais ou Cai 2 MOEA LARK VAIN CARS ALPE PEAS Be ©. Valentine Kirby, M.A. 
BOTA TIBD | ch GIR day ioc iZat be ure eat ak AEG Ctl ha tea al Co une amr Orton Lowe, BS. 
Foreign Gn guaeee es see weg welne eke Pa adit G. C. L. Riemer, M.A., Ph.D. 
RFGOCHADINY Ace acre ua UME ae as Nid Wa OK hcg ieee Erna Grassmuck, B.S. in Ed. 
High (School Inspections iy Caieideiels + been asec: weeeis James G. Pentz, B.A,, M.A. 
unior:' Ach: Schoolay ic Bievelscati omvelbrs cy aete Win tial ates wes pains M. ST aK: B.A., M.A. 
Mathems ies, ics oi sie ere wi ate loc nntare Sah Oh a eLe eke th ba anak care tpi edly aeenie HBG Krant J. A, Foberg, B.S. 
EBT Ue SR crate POS AR ras ta a cae a Se . -Hollis Dann, Mus.D. 
School Libraries, oi5c:6.iesjes Heda eTy ibe Sis bie Mh tos ara a evel Sera Wut oon Adeline B. Zachert 
Social Studies, ........ RS oR EE Marea Se. Edge Na J. Lynn Barnard, B.S., Ph.D. 


Speech Improvement, ........ Re REE GA ESS RUC EVAL EA Si aR AE Helen M. Peppard, B.A. 


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MATH 


-COMMONWEALTH- OF PENNSYLVANIA - - 


e DEPARTMENT OF PUBLIC INSTRUCTION 


COURSE oF STUDY 
IN © 


MATHEMATICS 


YEARS VIL-XII 


HARRISBURG, PENNSYLVANIA. 
J. L. L. KUHN, PRINTER TO THE COMMONWEALTH 
1983 


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2 ei SYLLABUS IN MATHEMATICS FOR HIGH 
a 
on SCHOOLS 
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Seal GENERAL STATEMENT 
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X Algebra and geometry originally college subjects. A study of the 
} mathematical courses and entrance requirements of early American 


, colleges shows that algebra and geometry were originally taught in 
< the colleges, and that later both subjects were dropped down bodily 
into the high schools and academies, becoming college preparatory 
subjects. The high schools and academies then taught identically 
the same courses that the colleges formerly had given, using precisely 
the same textbooks. The mathematics curriculum for the secondary 
school thus came to be organized from the standpoint of the adult 
mind, resulting in a formal, logical classification of subject matter. 


‘ Present organization is traditional. Our high school mathematics 
r> . courses have traditionally been organized so as to keep each subject 
> separate from the others: algebra first, then plane geometry, and so 
; on. This organization delays the pupil’s introduction to very signi- 
‘ ficant fields of elementary mathematics, such as numerical trigo- 
.= nometry, the use of logarithms, and other tables, graphical methods, 
and elementary notions of statistics, and denies him the stimulus and 

satisfaction that comes with real applications of elementary mathe- 
e matics to the solution of problems. 

The conventional organization, furthermore, carries the study of 
the technique and content of these isolated subjects to a point of 
completeness beyond that which can be justified for immature learn- 
ers, 


Recommendations growing out of recent studies. The world-wide 
movement for the reorganization of mathematics courses and teaching 
that has been in progress for a number of years has resulted (in our 
ewn country) in reports and recommendations by such important 
bodies as the National Committee on Mathematical Requirements, 
acting under the auspices of the Mathematical Association of 
America, and the Commission on the Reorganization of Secondary 
Ednuzation, appointed by the National Education Association. The 
recommendations of these two bodies agree that it is necessary to 

a give up “logical” arrangement of subject matter, especially for intro- 
ductory work, and to find instead an organization based upon the 
snecessful attack of projects and problems in connection with which 
the pupils already have hoth knowledge and potential interest. They 


4 


likewise agree in advocating the inclusion of arithmetic, algebra, 
and geometry in the course of study extending through the ninth 
school year. 


General mathematics. Large numbers of pupils leave the high 
school by the close of the ninth school year. For these pupils train- 
ing and instruction in “general” or “composite” mathematics, com- 
prising the fundamental ideas of the various branches of elementary 
mathematics, are more widely applicable to the experiences of every- 
day life than are the training and instruction limited to algebra 
alone. For the pupils who will remain in school and continue the 
study of mathematics, “general” or “composite” courses provide the 
best foundation for the study of mathematics in the second and later 
years. 


Fundamental principles. It is agreed that the emphasis should 
be placed throughout on the immediate values of the mathematics 
subjects studied—not preparation for the study of more mathematics 
at a remote date in the future, but the acquisition of knowledge and 
training that will enable the pupil to understand the relations of 
quantity and space entering into his daily life now. The course 
should be so planned that if the pupil is compelled to drop out of 
school at any time before the end of the term. the subjects studied 
up to that time shall have given him the most valuable mathematical 
training that he could receive. So planned, the course will likewise 
be the most natural and effective introduction to mathematics for 
those pupils who will continue the study of mathematics. 


RESPONSIBILITY OF THE TEACHER 


Knowledge of subject. It has been said that mathematics teachers 
“teach the textbook” to a greater extent than any other group of 
teachers. A reorganization of materials and methods of instruction 
can become effective only as the teachers assume responsibility for 
the choice and organization of the subject matter and presentation. 
The teacher must know his subject thoroughlv, and be able to adapt 
materials and methods to the varying needs of pupils. No method 
of presentation, and no particular choice of subject matter, can be 
permanently valid. 


Choice and organization of subject matter. In the absence of text- 
books embodying general agreement as to the choice and organization 
of the materials of instruction recommended for courses in composite 
mathematics, the teacher must assume responsibility for .choosing 
the needed arithmetic material from any good arithmetic text, the 
algebra material from any good algebra text, and the geometry mater. 
ial from any good geometry text. To teach high school mathematics 
so as to give the pupil the best mathematical training he is fitted to 


5 


receive—resulting in the broad acquaintance with mathematics as a 
whole that will give him an understanding of the relations of quantity 
and space growing out of his every-day environment, and that will 
enable him to decide wisely whether or not to continue the study of 
mathematics—calls for real skill in the teacher. To meet this de- 
mand is the inspiring task of the mathematics teacher of today. 


Professional contacts. No teacher of mathematics can do his work 
adequately unless he keeps in touch with the movement now going on 
everywhere in the interest of reform and improvement in the teaching 
of mathematics. This demands interest and participation in the work 
of such organizations as the Mathematical Association of America, 
the Association of Mathematics Teachers of the Middle States and 
Maryland, and the National Council of Teachers of Mathematics. 
The thoughtful reading of professional literature, including books 
and periodicals, is absolutely essential to successful teaching of high 
school mathematics. In particular, it is assumed that teachers will 
become familiar with standard practice material and with standard 
tests in high school mathematics subjects, and that they will employ 
both types of material in classroom practice. 


MATHEMATICS OF THE JUNIOR HIGH SCHOOL 


Fundamental principles. In the junior high school, comprising 
years seven, eight, and nine, the mathematics work should give the 
pupil as broad an outlook on the whole field of mathematics as he is 
able to comprehend, to the end that he may try out his capacities and 
aptitudes and learn whether or not he wishes to go more deeply into 
the study of mathematical topics and to take up the study of scienti- 
fic and technical subjects that demand extensive mathematical equip- 
ment. In addition, the mathematics work of these years should pro- 
vide the acquaintance with fundamental notions of elementary mathe- 
matics that has come to be regarded as essential to intelligent citi- 
zenship in the world of today. 


Content. The most authoritative opinions available, as expressed 
in the report and recommendations of the National Committee on 
Mathematieal Requirements, agree that the mathematics course in 
the junior high school should comprise arithmetic, the elementary 
notions of intuitive geometry and of algebra, and numerical trigo- 
nometry. The following paragraphs present, in outline, the topics 
that may appropriately be included under each of these headings. 


CONTENT OF JUNIOR HIGH SCHOOL MATHEMATICS 
Arithmetic 


A Practice in the fundamental operations applied to integers and 
fractions, common and decimal, should be continued until standard 


6 


proficiency is attained. When pupils attain standards, they should 
be excused from regular drill work. 

1 In working with fractions, the emphasis is to be placed on 

simple fractions: such as 4, %, %, 14, 34, %, 1%, et cetera. 

2 The decimal equivalents of fractions most commonly used 

should be fixed in mind, and the process of reducing any 

fraction to a decimal should be automatized. Rapid drill in 

one-step operations should be used for speed and accuracy. 


Material for this purpose may be of the following type: 


Ie SOLU eee 

207 OL OU a 

12 = What decimal part of 60? 
12 = What fractional part of 60? 


> The accurate placing of the decimal point in the process of 

division should be made automatic by fixing the habit of 

writing the quotient over the dividend, and of making the 
divisor an integer by multiplying, if necessary, both divid- 
end and divisor by the appropriate power of ten. 

4 Simple short cuts in multiplication and division: such as re- 
placing multiplication by 25 by multiplication by 100, and 
division of this result by 4. 

In the solution of problems, care is to be taken to pass froin 
the solution of the particular concrete problem to tue form- 
ulation of a general rule. 

The sequence, starting with the problem stated in 
numbers, through the problems stated in letters, the rule, 
the formula, and the problem without numbers, may be 
illustrated by the following series. The differences in 
degree of difficulty vary from step to step, and care must 
be taken that the pupils are not unduly hurried in the 
generalizations. 

a The length of a sidewalk is 50 ft., and its width is 3 ft. 
What is its area? 

b The length of a sidewalk is / ft. and its width w; 

What is A (the area) in terms of 1 and w? 

e State the rule for finding the area when length and 
and width are known. 

d. Write the formula for A, when ] and w are given. 
Write the formula for 1 when A and w are given. 

e The length and width of a sidéwalk are given—How may 
I figure the cost of laying it, if the cost of one square 
yard is given? 

6 In general, problems involving long computations should not 

be done in class time. 


Cr 


t 


B Tables of weights and measures in common use. United States 
money, avoirdupois weight, dry measure, liquid measure linear 
measure, square measure, cubic measure, time measure. 


C Percentage 


1 Fixing the equivalence of meanings of the symbols for frac- 
tion, decimal, and per cent. Graphs and diagrams to be 
used as helps in visualizing these ‘equivalences. 


2 Interchanging common fractions and percentages, finding 


any per cent of a number, finding what per cent one number 
is of another, finding a number when a certain per cent of it 
is known; such applications of percentage as come within 
the pupil’s experience. 

3 Much material must be provided for quick mental work, of 

the following type: 
50% of 200=— ? 
terol ou == + 
40 is what per cent of 200? 

Application of percentage 

a Interest 

The general method of figuring interest should be 
taught, and also the use of interest tables. Thrift and 
interest, studied in connection with savings accounts. Com- 
pound interest. 

b Profit and loss. 

Stress the need of care in choosing the base on which 
to compute the per cent of prifit or loss. Usually this 
base is the cost: sometimes the sel ing price is used. 
¢ Commission. 

To be computed on the amount of purchases or sales 

handled by the agent. 

d Discount. 
Reasons for “2% off for cash” 
Successive discounts. 

5 The following sequence illustrates desirable work to be done 
under the applications of percentage. Care must be taken 
to recognize the increase in difficulty as soon as general- 
ization is undertaken, and to anticipate these difficulties. 

a <A house was bought for $5600, and sold for $6150. What 
was the per cent of gain? 

b A house was bought for P dollars, and sold for S dollars: 
What was the per cent of gain? 

ec State the rule for finding the per cent of gain or loss 
when an article is sold. 


_ 


8 


d Write the formula for gain per cent, in terms of P 
(purchase price) and S (Selling price), when S is 
greater than P. 

Write the formula for a loss per cent. 

e How shall I find the gain per cent in a transaction which 
involves buying goods, paying cartage on them to the 
store, and then selling them? 


® 


D Graphic- Representation 


Making and interpreting line, bar, and circle graphs and apply- 
ing them wherever they can be used to advantage: care to 
be taken in making and interpreting pictorial representa- 
tions, because of the danger of misrepresenting facts when 
more than one dimension enters into the graph. Much 
inaterial for the application of graphical methods will be 
found in the field of social studies and geography. 

Cooperation between the teachers of these classes and the 
mathematics teacher is urgently recommended. 


E Business Practice 


1 JXeeping accounts: family budgets, personal budgets, family 


accounts, personal accounts. 


2 Arithmetic of the community: insurance 

Discussion of the need for insurance. 

a Fire insurance 

b Insurance of other kinds: plate glass, automobile, hail- 
storm, et cetera. 

ec Life and accident insurance. 

Problems on insurance to involve only the simple applica- 
tions. 


5 Arithmetic of civic life: taxes. 

a City taxes. 
Sources of expense in local governments, and ways 
of levying taxes. 

b National revenue. 
Sources of expense in national governments, and ways 
of raising revenue: customs and duties, income tax, 
inheritance tax, luxury tax, et cetera. 


4 Arithmetic of banking. 
a Different kinds of banks. 
b Savings account and checking account: deposit slips 
for each. 


¢ Writing and endorsing a check. 

d Making out withdrawal slip for savings account. 

e Borrowing money: promissory note. 

f Transmitting money to distant places. 

5 Arithmetic of investment. 

a Buying and selling real estate. 

b Stocks and bonds: fix clearly the difference between 
these two as investments. Use market reports of news- 
papers for problem material. Is a bond promising 10% 
return always better than one promising 6%?  Import- 
ance of dealing with reliable agents: consulting with 
the local banker. 

ec Commission and brokerage applied to real estate and 
stocks and bonds. 


Intuitive Geometry 


This subject should acquaint the pupil in a simple and interesting 
way with the most important geometric forms and their applications, 
through directed observation and experiment. 


A Simple Geometrical Figures. Familiarity with properties of 
equilateral triangle, 30°—60° right triangle, isosceles right triangle, 
circle, square, regular hexagon: symmetry, axial and central; know- 
ledge of the facts concerning the sum of the angles in a triangle; the 
pythagorean theorem; simple solids. such as a cube, pyramid, cone, 
prism, sphere. In connection with this work, accustom pupils from 
the beginning to using “circle” to mean the closed curve, and “poly- 
gon” to mean the closed broken line. “Area of the circle” and “area 
of the polygon” are to be taken to mean the areas enclosed by the 
lines. 


B Simple Geometrical Drawing. Use of T-square, triangles, pro- 
tractor and compasses: constructing perpendicular bisectors of lines, 
bisectors of angles, parallel lines; constructing triangles from given 
data; regular polygons, simple designs for ornament, et cetera. Use 
of squared paper. 


C Direct Measurements. Use of linear scales and protractor. 
Appreciation of the fact that these measurements are approximations, 
and development of judgment in the use of such data in computation: 
particular attention to be given to the number of figures to be re- 
tained in computations with approximate data. 


D_ Indirect Measurements. Making simple drawings to scale from 
actual measurements made by the pupil, and using them to get data 
not secured directly. Such applications to measuring heights and 
distances as are given in the Boy Scouts Handbook. 


10 


E Informal treatment of the idea of similarity. Drawing to 
scale: plans, working drawings, maps; use of squared paper; simple 
applications of proportion. 


F Mensuration. Area of square, rectangle, parallelogram, tri- 
angle, and trapezoid; length and area of circle; surfaces and volumes 
of solids of corresponding importance. These facts to be established 
in an experimental way, so far as possible by the pupil’s own activity. 
In the mensuration of the circle, use the value = for 7, avoiding 
the use of 3.1416, except in the rare cases where it is warranted by 
the exactness of the measurements. 


G Geometry of Appreciation. Geometrical forms in nature and 
art; in flowers, fruit, leaves, and animal forms; in architecture, manu- 
facture, and industry. 

The pupil’s active participation in discovering and describing the 
geometrical forms here mentioned is essential. The work in intui- 
tive geometry has general educational value in every day affairs, and 
serves as an introduction to demonstrative geometry. Along with 
the acquisition of fundamental ideas concerning the size, shape, and 
position of geometric forms in the plane and in space, the pupil 
should begin to make inferences and draw valid conclusions from 
facts discovered experimentally. 


Algebra 


A The formula. The construction of formulas by the pupil, as 
the outcome of work in mensuration. The following may serve as 
types: 

D=,r. t. (Distance in terms of rate and time) 
A=1/2b.h. (Area of triangle) 
—214+2h (Perimeter of rectangle) 
= bD.w.h. (Volume of rectangular solid) 
O— rd .(Circle) 

At a later stage, the pupils will be encouraged to bring to class 
formulas they have encountered in their reading, and to evaluate and 
explain them. 

Care should be taken that the material used is not too far separated 
from the pupil’s own experience. 


B Graphic representation. Statistical data from the field of geo- 
graphy and the social studies supply the material for the first use of 
graphs. 

Probably the first point at which the graph will be appreciated by 
the pupil in the study of algebra will be in connection with the study 
of the number-pairs that satisfy an equation like 2x — 3y = 46. and 
their determination of the points that constitute the graph of the 
equation. 


11 


‘he graph is to be used throughout the study of high school mathe- 
matics and is not to be taught as a separate topic. 

Whenever a diagram can be made to help in understanding a pro- 
blem or discussion, the pupil should be encouraged to make use of it, 
until the making of such graphs or diagrams becomes a fixed habit. 


C Positive and Negative Numbers. Their meaning and use as 
expressing both magnitude and one of two opposite directions or 
senses. 

Their graphic representation in connection with the representation 
of real numbers by the points on a line, emphasizing the fact that of 
two given numbers, that one is the greater whose point-representa- 
tion lies farther to the right on the axis of real numbers. 

The fundamental operations applied to them. 

Avoid any attempt to prove the laws of signs for the multiplication 
of signed numbers: these laws to be given as definitions. 


D The Equation. The introductory problems leading to the use 
~ of the equation must be very simple so that stress can be laid upon 
the mastery of the brief and systematic treatment of numerical re- 
lations that will later enable the pupil to solve problems otherwise 
beyond his power. 

At first the equation will involve only positive numbers. 

The negative numbers that appear later should be treated in a very 
simple manner, by adding an amount sufficient to make up the amount 
subtracted. Thus, in the problem: “The sum of two numbers is 94, 
and their difference is 38: find the numbers.” 

(94-x) ~x == 88 
94-2x = 88 

The —2x is disposed of by adding 2x to both sides, making the 

equation read 


94 — 38 + 2x, 
and the 38 on the right side disappears upon substracting 38 from 
both sides, getting 
56 = 2x, 
whence x = 28 

It is well to accustom the pupil early in his study of algebra to 
look upon the form 56 —2x as being just as good as the form 2x =—56. 

Throughout the course, stress will be laid upon the solution of 
problems, involving the translation of the verbal statement into alge- 
braic language. 

Equations in two variables will be restricted to sets of linear 
equations at first. In a first course, elimination by addition and sub- 
traction to be the only method employed: solution to be checked by 
graphs. The solution of the quadratic equation to be restricted to 
the “pure” quadratics in one unknown. 


(a2 


A simple treatment of proportion, including various applications 
of ratio and proportion to the problems of ordinary experience. Use 
Ty eau 
the fractional notation --==* discarding the archaic “dot” 
a ¢ 
notation. © Discard the terms “antecedent” and “consequent,” using 
“numerator” and “denominator” instead. 


Algebraic Lechnique 


A, The Four Fundamental Operations. Their treatment should 
include the representation of numerical relations by means of al- 
gebraic symbols, and the translation of symbolic expressions into 
words. 

Multiplication and division should rarely involve anything beyond 
monomial and binomial multipliers, divisors, and quotients. Com- 
plications never met in the practical applications of algebra (as, 
for instance, nests of parentheses) should be avoided. 

The treatment of literal equations should be restricted to the ma- 
terial necessary for attaining facility in manipulating formulas. 

The solutions of problems should involve the verification of the re- 
sults obtained. The feeling of self-confidence and certainty that his 
work is correct is a valuable outcome from the pupil’s study of math- 
ematics. 


B- Factoring. The only cases that need to be treated are: 
1 Monomial factors, as in ax-+ay + az. 

2 The difference of two squares. 

3 The square of a binomial. 


C Fractions. The four fundamental operations should be ap- 
plied to simple cases, constantly through the course, avoiding com- 
plicated forms that are never met with in practice. 


Shanging fractions to equivalent fractions having different de- 
nominators should be made to depend on the pupil’s recognition of 
the need of multiplying the old numerator by the same factor that 
the old denominator was multiplied by in obtaining the new denom- 

3a ? 


inator. Thus, — —=-—— _ will be completed by answering the ques- 
4b 20be 


tions “What was the denominator 4b multiplied by to make the new 
denominator 20be?” and “What, then, must be done to the old numer- 
ator 3a?” 


“Clearing an equation of fractions” should be rationalized for the 
pupil, so that he knows that the process involves multiplying every 
term by a multiplier that will cause the fractions to disappear. 
Separate treatment of greatest common divisor, and lowest cor 


13 
mon denominator, isolated from the treatment of equations involv- 
ing fractions, is inadvisable. 
When such an equation as 


he es 4 (x~=-2 Ux — J 


ge eS 


5 10 8 
is to be solved, call first for the method of getting rid of the largest 
denominator. “By multip'ying through by 10.”’—This gives the 
result 


2 5 
Mi3x- 4) | Wi7x - 2) .~ 10(9x - 1) 
54 AT oh x, 


leaving a denominator 4, which may be made to disappear by multi- 
plying through by 4, giving 


24x — 32 + 28x — 8 = 45x - 5, 
from which 

52x — 40 = 45x — 5 
and 

nese ey 


By the time the majority of the class has acquired facility in this 
process of solution, some member is fairly sure to suggest replacing 
the separate multiplications by 10 and 4, by a single multiplication 
by 40. 


Then the class may be set the task of determining the single mul- 
tipliers that will get rid of all the denominators at one time, in 
similar equations. 


Complex fractions should be restricted to such as are not more 
difficult than 
a ec 


sd 
b ad 
m p —— 


n q 
D Exponents and Radicals. The treatment of these topics should 
be confined to the simplest material needed for the treatment of 
formulas, 
Care should be taken to make it clear that the symbol Va (a 
representing a positive number) means only the positive square 


is 
root, and that \’/a means only the principal ath root. It is 
therefore improper to write x = \/ 3 as the complete solution of 
x? — 3 = 0, of which the result should be written x = + V/3. 


14 


. 
In connection with the work with the pythagorean theorem, it 


will be necessary to teach a process for finding the square roots of | 
numbers. For this purpose, the following process is recommended: 


Required, to find the value of \/5486. : 
The pupil knows that 70? = 4900 
and that SO? = 6400 


and consequently that 1/5486 lies between 70 and 80, and is nearer 70 
than 80. When required to estimate its value, he may say 72. The 
test of 72 as the square root is made by dividing it into the given 
number. The result, 76.2, indicates that 72 is not the true square 
root, and further that the value of the square root is larger than 72, 
and smaller than 76.2. The mean value of these two numbers, 74.1, 
may then be taken as the required value of 1/5486. If a greater 
degree of accuracy is required, the process of checking by division, 
and correcting by taking the mean value, may be repeated. The ad- 
vantage of this process is that it compels the pupils to form correct 
conceptions of the meaning of square root, and that its simplicity 
ensures retention. Ata later time, the traditional method based on 
the expansion of (a -+- b)* may be taught, if considered desirable. 


Numerical Trigonometry 

A Definitions of sine, cosine and tangent, arrived at in an ex- 
perimental way, through measurements and. tabulations made by 
the pupil. Constructing angles when values of the functions are 
given. 

B The wse of the functions in solving simple problems in heights 
and distances. In this connection it is highly important that the 
pupils themselves make the measurements needed—that field-work 
of a very simple kind be required. Instruments constructed by the 
class are better for this purpose than surveyors’ instruments. 
Measurements and constructions with steel tape; heights by 
“shadow” method and by measurements of distance and angle of 
elevation; areas of irregular plots of ground. 


C The construction of very simple tables of the functions by the 
pupils, andthe use of 3 or 4 place printed tables of natural values 
of the functions. 

In the work done on this topic, attention will be concentrated 
upon, and limited to, the simplest fundamental notions. 


Problems 


Must be real to the pupil. Throughout the course in high school 
mathematics, the solution of problems must be given major emphasis. 
So far as the pupil’s maturity and knowledge of science and industry 
permit, “practical problems” should be freely introduced in high 
school mathematics teaching. Care must be taken to ensure that _ 


15 


these problems are real to the pupil, that they connect with his or- 
dinary thoughts, and lie within the world of his interest and ex- 
perience. 


Computation. In the solution of problems, opportunity will con- 
tinually be presented for arithmetical and computational work. The 
notion that algebra is an extension of arithmetic should be empha- 
sized both in numerical work and in explaining algebraic principles. 
Computations and the solutions of problems should habitually be 
checked. The use of approximate data in computation presents 
opportunities to stress the need for exercising Common sense and 
judgment especially with regard to the number of significant figures 
retained in the results of computations. Pupils should gradually be 
accustomed to the use of such tables as squares and square roots, 
cubes and cube roots, trigonometric functions, interest, and the like, 
to facilitate computation. 


Argebra Topics to be Omitted 


The following topics should be omitted from a first course in high 


school mathematics: " 

Drill in algebraic technique designed merely to secure faclity and skill 
in manipulation, apart from the acquisition of power to attack signifi- 
eant problems. 

Cases in factoring other than those I'sted. 

Highest common factor and lowest common multiple as separate top‘es. 

The theorems on proportion relating to alternation, inversion, composi- 
tion, and division. 

Literal equations, except such as appear in connection with work on for- 
mulas. 

Radicals, except as indicated in section D, page 13. 

Square roots of polynomials. ¥ 

Cube root. 

Theory of exponents. 

Simultaneous equations in more than two unknowns. 

Pairs of simultaneous quadratic equations. 

The theory of quadratics (remainder and factor theorems, ete.) 

All equations of degree higher than the second. 

The binomial theorem. 

Arithmetic and geometric progressions. 

Theory of imaginary and complex numbers. 

Radical equations, except such as arise in dealing with elementary 
formulas. 


THE COURSE OF STUDY IN MATHEMATICS FOR THE JUNIOR 
HIGH SCHOOL 


The topics to be included in the Course of Study for the Junior 
High School may be arranged in the following manner: 
Seventh year: Arithmetic, particularly as applied to the 
home, to industry, and to the other subjects 


16 
in the school curriculum. Intuitive geom- 
etry. 

Kighth year: Algebra; arithmetic, particularly its social 
and commercial applications. Such geom- 
etry as grows naturally out of discussions of 
the size and shape of figures. 

Ninth year: Algebra; numerical trigonometry; where 
possible, and introduction to demonstrative 

7 geometry, with the aim limited to learning 
the meaning of “demonstration.” 


NINTH YEAR MATHEMATICS IN THE FOUR-YEAR HIGH 
SCHOOL 


Purpose and content. In four-year high schools, the ninth year’s 
work in mathematics should provide the pupil with as broad a foun- 
dation of mathematical training as possible. In particular, the 
course should include algebra, numerical trigonometry, and geometry, 
with at least an indication of the nature of a geometric demonstration. 


Time allotment. In such schools, it igerecommended that about 
two-thirds of the ninth year be devoted to algebra and numerical 
trigonometry, and about one-third to geometry. The outline of 
subject matter for the junior high school mathematics will be of use 
in connection with the type of school here discussed, provided proper 
allowance is made for the greater maturity of the pupils, and for 
the fact that the mensuration work done in the seventh and eighth 
years of the elementary school has furnished a considerable body of 
geometric training. 


Guidance in electing mathematics. Provision must be made 
through the guidance program of the school, or otherwise, to ensure 
that all pupils are properly informed of the vital importance of 
mathematics in many lines of endeavor in adult life, and to see to it 
that pupils are properly advised when they decide whether or net 
to continue the study of mathematics beyond the ninth year. 


Transition from algebra to general mathematics. Algebra will 
continue for some time to be the mathematics taught in the ninth 
year in many high schools. This is as it should be, in view of the 
fact that radical departure from customary practice without 
preparation would be unwise. However, in view of the advantage 
of the course in mathematics for the ninth year comprising the 
fundamental notions of algebra, intuitive geometry and numerical 
trigonometry, it is confidently expected that gradually ninth year 
algebra will be replaced by a year of composite mathematics. 
There will be a period of transition, in which the material of in- 
struction will be organized by teachers, eliminating a considerable 


body of algebra material that has been customarily given, and 
bringing in new material from the geometry side. The final result 
of this process should be a year’s work involving the fundamental 
notions of the elementary branches of mathematics that will con- 
tribute most largely to the pupil’s training for intelligent citizenship, 
as well as furnish the best foundation for further work in mathe- 
maties. 


COURSE IN ALGEBRA COVERING NINTH YEAR 


Ninth Year Algebra. The following outline is designed to serve 
the needs of schools in which the ninth year is devoted to the study 
of algebra. 

The Formula. Material leading to the formula will have been 
met in the pupil’s work in arithmetic, in connection with such topics 
as interest, and mensuration. This experience with the formula 
should be reviewed, and extended. 

Special care should be taken to see that the pupils understand the 
algebraic language into which the verbal statements are translated. 

The dependence of one quantity upon another, as expressed by the 
formula, should be clearly understood. Such questions as the follow- 
ing are appropriate: 

Given the formula V=1/3zr*h. (a) If V is to be kept constant 
how must h*change when r increases? (b) If r is kept 
fixed, how will V change when h is doubled? 

In the science class, there will be occasion to study such formulas 
as: 


1D) 


Ohm’s Law: I Sy 
Law of machines:.P K Pd=W xX Wd 


Thermometer : C==—(F—32) 


Such formulas as these should be taken up in the algebra class- 
room as one phase of codperation between the teachers of science 
and mathematics. 


Graphic Representation. This should not be considered an_iso- 
lated topic, but should be used throughout the year’s work, whenever 
helpful, as an illustrative and interpretative instrument. 

Value coérdination may be effected by coédrdination between 
the mathematics teacher and the science teacher, in making gravlis 
for such topics as the following: 

Production of coal, petroleum, etc., in the United States. 
Distribution of elements in earth’s crust. 


18 


Range of wave-lengths of heat, sound, wireless, etc. 
Cost of gas used in various appliances. 

Food values of various substances. 

Hours of light and darkness through the year. 
Health statistics. 

Temperature graphs. 


Positive and Negative Numbers. Their meaning and use as 
expressing both magnitude and one of two opposite directions or 
senses. - 


Their graphic representation in connection with the representa- 
tion of real numbers by the points on a line, emphasizing the fact 
that of two given numbers, that one is the greater whose point- 
representation lies farther to the right on the axis of real numbers. 


The fundamental operations applied to them. No attempt should 
be made to prove the laws of signs for the multiplication and division 
of signed numbers; these laws should be given as definitions. 


The Equation. (a) Linear equations in one unknown—setting up 
such equations as translations of verbally given problems, and solv- 
ing them. There should be much drill at first on very simple types of 
equations. 


The use of “transpose” as a technical term should be postponed 
until late in the year, using at first the phrase “add to both sides” 
or “subtract from both sides.” The habit of combining terms in 
both members of the equation should be established as the first step 
in solving an equation. 


Thus the equation: 
73. hl Be moe > es eae pen alt Pda 
should first be simplified to 
—x-+8& = —4x-++9 
and then, by adding 4x to both sides, and subtracting 8 from both 
sides, we have 


ox == 1 
(b) Simple cases of quadratic equations that arise in solving prob- 
lems and in handling formulas. 
(c) Sets of equations in two unknowns, limited to pairs of linear 
equations. 


(d) Applications of ratio and proportion to simple cases of simi- 
larity and other problems of ordinary life. The proportion should al- 
ways be written as an equation between two fractions, and in solv- 
ing the proportion, it should be treated as an equation. The terms 
“means,” “extremes,” “antecedent,” “consequent,” should be discarded 
in favor of “numerator” and denominator.” 


19 


Algebraic Technique. The Four Fundamental Operations. Their 
connection with the processes of arithmetic should be made clear. 

Multiplication and division should rarely involve multipliers or 
divisors of more than two terms. 

“Nests” of parentheses should not be treated, because of their 
rare occurrence in practical applications. 

Literal equations should be treated only to the extent necessary 
for manipulating formulas. 

The habit of verifying solutions should be established. 


Factoring. The only cases that need to be treated are: 
1. Monomial factors. 
2. The difference of two squares. 
3. The square of a binomial. 


Skill acquired in factoring other cases—as the sum of two cubes, 
for instance—is wasted because no opportunity is ever presented 
for its use in applications of mathematics to real situations. 


Fractions. The four fundamental operations should be applied 
only to simple cases, throughout the course, avoiding complicated 
forms that are never met with in practice. 

The use of “cancel” as a technical term should be postponed until 
late in the year, using instead the phrase “divide numerator and de- 
nominator by.” 

ax x 
Errors of the type— = — should be forestalled by in- 
ayt2 yz 
sistence upon division of both terms of the fraction in this simplifi- 
cation. 

Separate treatment of “least common denominator” and “highest 
common divisor’ should not be given. “Clearing an equation of 
fractions” should be rationalized as indicated on page 13. 

Somplex fractions should be restricted to such as are not more 
difficult than 


a C 

psa 
m Pp 
nm og 


Haponents and Radicals. The work done on exponents and radi- 
cals should be confined to the simplest material required for the 
treatment of formulas. 

Proofs of the laws for positive integral exponents should be in- 
cluded. Care should be taken in this connection to ensure under- 
standing of the facts dealt with, by illustrations from the arithmetic 


20 \: 
side. Thus appreciation of the truth of 2? x 2? = 2°, by translat- 
ing into 4 x 8=382, should precede learning x* x°==x*T?, 


The meaning and use of fractional and negative exponents should | 


‘be considered in connection with handling formulas. With some 
classes, it will be possible to include an elementary discussion of 
logarithms and the slide rule. 

The consideration of radicals should be confined to transforma- 
tions of the types: 

- fab = a/b and \/a/b =1/b\/ab. 

A process for finding the square root of a number should be taught 
(see page 14), but time should never be given to extracting roots of 
-algebraic polynomials. 

Problems. Most of the emphasis now frequently placed on formal 
exercises should be transferred to the solution of problems. Prob- 
lems should be “practical” so far as the maturity of the pupil per- 
mits. 

Problems should always be “real” to the pupil, should connect 
with his ordinary thought, and be within the world of his interests 
and experience. 

A conscious effect should be made, in the selection of problems, 
to correlate the work in mathematics with the other courses in the 
curriculum, particularly with the courses in science. 


Numerical Trigonometry. In view of the fact that this sub- 
ject has been recommended for inclusion in the College Entrance 
Examination Board Examination in Elementary Algebra (Part I), 
Algebra to Quadratics, it probably will be considered desirable to 
include numerical trigonometry with algebra in all schools devot- 
ing the whole ninth year to algebra. 

An outline will be found on page 14.’ 


TENTH YEAR MATHEMATICS 


Election and Guidance. It is assumed that the ninth year’s work 
in mathematics is the final year’s work in required mathematics. 
Later work in this subject will be elective, and elections will be made 
under the guidance of competent teacher-advisers. This assump- 
tion of adjustment to the individual needs and aptitudes of pupils 
is increasingly justified as proper recognition is given to the im- 
portance of guidance activities in the high school. 


Demonstrative geometry. On the basis of the work done in pre- 
ceding years, whether.in junior high school or in the elementary 
school and ninth year in the four-year high school, the pupil who 
elects to continue the study of mathematics will be ready to take up 
the study of demonstrative geometry. In some schools the training 
in intuitive geometry already secured will enable the pupil to cover 


21 


the work in plane and solid geometry in one year. In this ‘case the 
greater part of this year will be devoted to plane geometry, and the 
iesser part (say, about one-third) to solid geometry. 


The work in demonstrative geometry should always be preceded 
by work in intutive geometry, such as is outlined on page 9 above 
If the pupils beginning demonstrative geometry have not already had 
this type of work, a short time (say, three weeks) should be devoted 
to such introductory work before proceeding to the strictly demon- 
strative type of work. 


, 

Assumptions and Theorems for Informal Treatment. This list 
contains propositions which may be assumed without proof 
(postulates), and theorems which it is permissible to treat informally. 
Some of these propositions will appear as definitions in certain 
methods of treatment. Moreover, teachers should feel free to re- 
quire formal proofs of some, if they desire to do so. The precise 
wording given is not essential, nor is the order in which the pro- 
positions are here listed: 


1 Through two distinct points it is possible to draw one 

| straight line, and only one. 

2 A line segment may be produced to any desired length. 

The shortest path between two points is the line segment 

joining them. 

4 One and only one perpendicular can be drawn through 

a given point to a given straight line. 

The shortest distance from a point to a line is the per- 

pendicular distance from the point to the line. 

6 From a given center and with a given radius one and 

only one circle can be described in a plane. 

7 A straight line intersects a circle in at most two points. 

S <Any figure may be moved from one place to another 
without changing its shape or size. 

9 All right angles are equal. 

10 If the sum of two adjacent angles equals a straight angle, 
their exterior sides form a straight line. 

11 Equal angles have equal complements and equal sup- 
plements. 

12 Vertical angles are equal. 

13. Two lines perpendicular to the same line are parallel. 

14 Through a given point not on a given straight line one 
straight line, and only one, can be drawn parallel to 
the given line. 

15 ‘Two lines parallel to the same line are parallel to each 
other. 


Ol 


22 


16 The area of a rectangle is equal to its base times its 
altitude. 


Pundamental Theorems. The following list of theorems in intend. 
ed to include those of sufficient importance to require demonstra- 
tion: 


1 Two triangles are congruent if* 

a Two sides and the included angle of one are equal, 
respectively, to two sides and the included angle of 
the other ; 

b Two angles and a side of one are equal, respectively, 
to two angles and the corresponding side of the 
other ; - 

c ‘The three sides of one are equal, respectively, to 
the three sides of the other. 

2 Two right triangles are congruent if the ee 
and one other side of one are equal, respectively, to 
the hypotenuse and another side of the other. 

3 If two sides of a triangle are equal the angles opposite 
these sides are equal; and conversely.** 

4 The locus of a point (in a plane) equidistant from two 
given points is the perpendicular bisector of the line 
segment joining them. 

The locus of a point equidistant from two given inter- 
secting lines is the pair of lines BERS the angles 
formed by these lines. 

6 When a transversal cuts two parallel lines, the alter- 

nate interior angles are equal; and conversely. 

7 The sum of the angles of a triangle is two right angles. 

S <A parallelogram is divided into congruent triangles by 
either diagonal. 

9 Any (convex) quadrilateral is a parallelogram, (a) if the 
opposite sides are equal; (b) if two sides are equal and 
parallel. 

10 Ifa series of parallel lines cut off equal segments on one 
transversal, they cut off equal segments on any trans- 
versal. 


Or 


* "Teachers should feel free to separate this theorem into three distinct theorems 
and to use other phraseology for any such proposition. For example in 1, 
“Two triangles are equalif..” ‘‘A triangle is determined by ....”, ete. Similarly, 
in 2, the statement might read: ‘Two right triangles are congruent if, besides 
the right angles, any two parts (not both angles) in the one are equal to correspond- 
ing parts of the other.” 


** Tt should be understood that the converse of a theorem need not be treated 
in connection with the theorem itself, it being sometimes better to treat it later. 
Furthermore, a converse may occasionally be aecepted as true in an elementary 
course, if the necessity for proof is made clear. The proof to be given later. 


a2 


tl 


14 


15 


18 


19 


23 


a The area of a parallelogram is equal to the base 

times the altitude. 

lb The area of a triangle is equal te one half the base 
times the altitude. 

¢ The area of a trapezoid is equal to half the sum of 
its bases times its altitude. 

d The area of a regular polygon is equal to half the 
product of its apothem and perimeter. 

a If a straight line is drawn through two sides of a 
triangle parallel to the third side, it divides these 
sides proportionally. 

b If a line divides two sides of a triangle proportion: 
ally, it is parallel to the third side. (Proof for 
commensurable case only.) 

c The segments cut off on two transversals by a series 
of parallels are proportional. 

Two triangles are similar if 

a They have three angles of one equal, respectively, to 
three angles of the other; 

b They have an angle of one equal to an angle of the 
other and the including sides are proportional; 

c Yheir sides are respectively proportional. 

If two chords intersect in a circle, the product of the 

segments of one is equal to the product of the segments 

of the other. 

The perimeters of two similar polygons have the same 

ratio as any two corresponding sides. 

Polygons are similar, if they can be decomposed into 

the same number of triangles, similar each to each and 

similarly placed; and conversely. 

The bisector of an (interior or exterior) angle of a tri- 

angle divides the opposite side (produced if necessary ) 

into segments proportional to the adjacent sides. 

The areas of two similar triangles (or polygons) are to 

each other as the squares of any two corresponding 

sides. 

In any right triangle the perpendicular from the vertex 

of the right angle on the hypotennse divides the tri- 

angle into two triangles each similar to the given tri- 
angle. 

In a right triangle tie square on the hypotenuse is 

equal to the sum of the squares on the other two sides. 

In the same circle or in equal circles, if two ares 

are equal, their central angles are equal; and converse- 

ly. 


24 


22 Jn any circle two angles at the center are proportional 
to their intercepted ares. (Proof for commensurable 
case only) , 

23 In the same circle or in equal circles, if two ares are’ 
equal their chords are equal; and conversely. 

24 a <A diameter perpendicular to a chord bisects the 

chord and the ares of the chord. 
b <A diameter which bisects a chord (that is not a dia- 
-» meter) is perpendicular to it. 

25 The tangent toa circle ata given point is perpendicular 
to the radius at that point; and conversely. 

26 In the same circle or in equal circles, equal chords are 
equally distant from the center; and conversely. 

27 An angle inscribed in a circle is equal to half the cen- 
tral angle having the same arc. 

28 Angles inscribed in the same segment are equal. 

29 If a circle is divided into equal ares, the chords of 
these arcs form a regular inscribed polygon and tan- 
gents to the points of division form a regular circum- 
scribed polygon. 

30 The circumference of a circle is equal to 2 7 r (informal 
proof only). 

31 *The area of a circle is equal to 7 r? (informal proof 
only) 


The treatment of the mensuration of the circle should be based on 
related theorems concerning regular polygons, but it should be in- 
formal as to the limiting processes involved. The aim should be an 
understanding of the concepts involved, so far as the capacity of the 
pupil permits. It is recommended that the theorems concerning 
the length of side of a polygon of 2n sides, in terms of the side of the 
polygon having n sides, for the inscribed and circumscribed polvgons, 
be omitted. The value of z should be given to four or five places, 
with the statement that its value to a greater degree of accuracy is 
found by methods of advanced mathematics. For cae Nags pur- 


9). 


22 
poses the value : should be used. 


FUNDAMENTAL CONSTRUCTIONS 


1 Bisect a line segment and draw the perpendicular bi- 
sector. 


*“The total number of theorems given in this list when separated (as will be 
found advantageous in teaching) including the converses indicated, is 52. 


ea) 


Subsidiary Propositions. 


Bisect an angle. 

Draw a perpendicular to a given line through a given 
point. 

Construct an angle equal to a given angle. 

Through a given point draw a straight lind parallel to 
a given straight line. 

Construct a triangle, given 

a Three sides 

b Two sides and the included angle 

ec Two angles and the included side | 
Divide a line segment into parts proportional to given 
segments. 

Given an are of a circle, find its center. 

Circumscribe a circle about a triangle. 

Inscribe a circle in a triangle. 

Construct a tangent to a circle through a given point. 
Construct the fourth proportional to three given line 
segments. 

Construct the mean proportional between two given line 
segments. 

Construct a triangle (polygon) similar to a given tri- 
angle (polygon). 

Construct a triangle equal to a given polygon. 
Inscribe a square in a circle. 

Inscribe a regular hexagon in a circle. 


exercises may be selected: 


1 


i) 


hj 


When two lines are cut by a transversal, if the corres- 
ponding angles are equal, or if the interior angles on 
the same side of the transversal are supplementary, 
the lines are parallel. 

When a transversal cuts two parallel lines, the corres- 
ponding angles are equal, and the interior angles on the 
same side of the transversal are supplementary. 

A line perpendicular to one of two parallels is perpen- 
dicular to the other also. 

If two angles have their sides respectively parallel or 
respectively perpendicular to each other, they are either 
equal or supplementary. 

Any exterior angle of a triangle is equal to the sum of 
the two opposite interior angles. 

The sum of the angles of a convex po’ygon of n sides is 
2(n—2) right angles 


The following list is intended to include 
material from which additional theorems, corollaries, originals, and 


9 


10 
11 


12 


ra) 


13 


14 


15 
16 


21 


a A 


*Such inequality theorems as these are of great importance in developing the 
notion of dependence or functionality in geometry. r 
in this “Subsidiary List of Propositions” should not imply that they are considered 
of less educational value than those in List II. Thev are placed here because they 
are not “fundamental” in the same sense that the theorems of List II are funda- 


mental, 


26 


In any parallelogram 
a the opposite sides are equal; 

b the opposite angles are equal; 
¢ the diagonals bisect each other. 

Any (convex) quadrilateral is a parallelogram, if 
a the opposite angles are equal; 

b the diagonals bisect each other. 

The medians of a triangle intersect in a point which is 
two-thirds of the distance from the vertex to the mid- 
point of the opposite side. 

The altitudes of a triangle meet in a point. 

The perpendicular bisectors of the sides of a triangle 

meet in a point. 

The bisectors of the angles of a triangle meet in a point. 
The tangents to a circle from an external point are 

equal. 

*a If two sides of a triangle are unequal, the greater 
side has the greater angle opposite it; and con- 
versely. 

b If two sides of one triangle are equal respectively 
to two sides of another triangle. but the included 
angle of the first is greater than the included angle 
of the second, then the third side of the first is 
greater than the third side of the second; and con- 
versely. 

c If two chords are unequal; the greater is at the less 
distance from the center, and conversely. 

d The greater of two minor arcs has the greater chord 
and conversely. 

An angle inscribed in a semi-circle is a right angle. 

Parallel lines, tangent to or cutting a circle, intercept 

equal arcs on the circle. 

An angle formed by a tangent and a chord of a circle is 

measured by half the intercepted are. 

An angle formed by two intersecting chords is measured 

by half the sum of the intercepted arcs. 

An angle formed by two secants, or by two tangents, 
to a circle, is measured by half the difference between 
the intercepted arcs. 

If from a point without a circle a secant and a tangent 
are drawn, the tangent is the mean proportional be- 
tween the whole secant and its external segment. 

Parallelograms, or triangles, of equal bases and alti- 


tudes are equal. 


The fact that they are placed 


27 


22 «©The perimeters of two regular polygons of the same 
number of sides, are to each other as their radii and 


also as their apothems, 


Solid Geometry A: combined with plane geometry in one year, 
For classes desiring to cover plane and solid geometry in one year, 
as indicated on page 21, the study of solid geometry should include 
the following: 

An informal treatment of the fundamental theorems dealing 
with the relations of lines and planes in space, based on the pupil’s 
experience, and familiarity with the substance of the usual definitions 
and theorems. 

An informal treatment of locus problems, emphasizing the acquisi- 
tion of power to visualize, describe and represent the figures dealt 
with, and putting relatively little emphasis upon formal proofs. 

Problems of measurement and calculation, stated in words or in 
terms df a figure, form the most important topic in the work on 
solid geometry. The pupil should be familiar with the mensuration 
formulas of plane geometry, with the use of the sine and tangent 
functions in solving right triangles, and in the course of the work 
should become familiar with the solid geometry formulas listed below. 

Similarity of solids should be carried far enough to make the 
pupil familiar with the facts that the volumes of similar solids are 
proportional to the cubes, and surfaces to the squares, of correspond- 
ing dimensions. 


Abilities to be acquired. Throughout the course, stress should be 
laid upon gaining ability to visualize the space relations of the 
figures dealt with, to describe them accurately in words, to represent 
them clearly by drawings, free-hand or made with instruments. 
The pupil should become able to use intelligently the terminology of 
the subject, including such terms as polyhedron, regular polyhedron, 
prism, cylinder, pyramid, cone, sphere; r:ght, regular and oblique, 
applied to solids; vertex, diagonal, face diagonal, lateral edge, 
altitude, slant height; zone of a sphere, lune, great circle, small 
circle; similar solids. 


Fundamental theorem in mensuration of volumes. Mensuration 
of volumes may be made to depend upon an informal treatment of 
the theorem: 

If two solids are included between two parallel planes, and are 
such that sections made by any plane parallel to the first two are 
equal in area, then the solids are equal in volume. 


Mensuration formulas. The following list of mensuration form- 
ulas should be included: 
Prism: lateral surface, yolume, 


28 


Right circular cylinder: lateral surface, -total surface, 
volume. 

Regular pyramid and right circular cone and their frus- 
tums: lateral surface and volume. 

Sphere: Surface, area of zone, area of lune of n degrees, 
volume. 


ELEVENTH YEAR MATHEMATICS 


Subject matter. The mathematics work of the eleventh year may 
be divided between advanced algebra and solid geometry, when the 
latter subject is not taken in the tenth year. The order in which 
these two topies is given is unimportant. When solid geometry is 
taken up in connection with plane geometry in the tenth year, the 
eleventh year will be divided between advanced algebra and trig- 
onometry. 


. 

For some groups of pupils, a course in shop mathematics may 
wisely be given in the eleventh year in place of the algebra-solid 
ceometry or algebra-trigonometry just indicated. Such a course is 
referred to briefly on page 37. 


Solid Geometry B: Covering One-half Year. For classes devoting 
a half year to solid geometry, the following list of propositions is 
recommended. The division into “fundamental” and “subsidiary” 
theorems will again call attention to the need for varying the 
emphasis given different portions of this material of instruction. 

In the following list the precise wording and the sequence are 
not considered: 


I. FUNDAMENTAL THEOREMS 


1. If two planes meet, they intersect in a straight line. 

Tf a line is perpendicular to each of two intersecting 

lines at their point of intersection it is perpendicular 

to the plane of the two lines. 

3. Every perpendicular to a given line at a given point 
lies in a plane perpendicular to the given line at the 
given point. “ayy 

4. Through a given point (internal or external) there can 
pass one and only one perpendicular to a plane. 

5. Two lines perpendicular to the same plane are parallel. 

6. If two lines are parallel, every plane containing one 
of the lines and only one is parallel to the other. 

7. Two planes perpendicular to the same line are parallel. 

If two parallel planes are cut by a third plane, the lines 

of intersection are parallel. 


bo 


N 


v. 


10, 


Le. 


12: 


16. 


17. 


1A) 


20. 


21. 


oF. 


19 3 


ou 


If two angles not in the same plane have their sides 
respectively parallel in the same sense, they are equal 
and their planes are parallel. 

If two planes are perpendicular to each other, a line 

drawn in one of them perpendicular to their inter- 

section is perpendicular to the other. 

If a line is perpendicular to a given plane, every plane 

Which contains this line is perpendicular to the given 

plane. 

if two intersecting planes are each perpendicular to a 

third plane, their intersection is also perpendicular 

to that plane. 

The sections of a prism made by parrallel planes cut- 
ting al the. lateral edges are congruent polygons. 
An oblique prism is equal to a right prism whose base 
is equal to a rght section of the oblique prism and 
whose altitude is equal to a lateral edge of the oblique 

prism. | 

The opposite faces of a parallelopiped are congruent. 

The plane passed through two diagonally opposite edges 

of a parallelopiped divides the parallelopiped into two 

equal triangular prisms. 

If a pyramid or a cone is cut by a plane parallel to 
the base: 

(a) The lateral edges and the altitude are divided 
proportionally ; 

(b) The section is a figure similar to the base; 

(c) The area of the section is to the area of the base 
as the square of its distance from the vertex is to 
the square of the altitude of the pyramid or cone. 

Two triangular pyramids having equal bases and equal 

altitudes are equal. 

All points on a circle of a sphere are equidistant from 

either pole of the circle. 

On any sphere a point which is at a quadrant’s distance 

from each of two other points not the extremities of a 

diameter is a pole of the great circle passing through 

these two points. 

If a plane is perpendicular to a radius at its extremity 

on a sphere, it is tangent to the sphere. 

A sphere can be inscribed in or circumscribed about any 

tetrahedron. 

If one spherical triangle is the polar of another, then 

reciprocally the second is the polar triangle of the 

first. 


4 


op Pas 


FTF 


if. 


Zo; 


4(). 


30 


In two polar triangles each angle of either is the sup- 
plement of the opposite side of the other. 
Two symmetric spherical triangles are equal. 


FUNDAMENTAL PROPOSITIONS IN MENSURATION 


The lateral area of a prism or a circular cylinder is 
equal to the product of a lateral edge or element re- 
spectively, by the perimeter of a right section. 

The volume of-a prism (including any parallelopiped) 
or of a circular cylinder is equal ‘to the product of its 
base by its altitude. 

The lateral arca of a regular pyramid or a right circular 
cone is equal to half the product of its slant height by 
the perimeter of its base. 

The volume of a pyramid or a cone is equal to one-third 
the product of its base by its altitude. 

The area of a sphere. 

The area of a spherical polygon. 

The volume of a sphere. 


Ill. SUBSIDIARY THEOREMS 


If from an external point a perpendicular and obliques 
are drawn to a plane, (a) the perpendicular is shorter 
than any oblique; (b) obliques meeting the plane at 
equal distances from the foot of the perpendicular are 
equal; (c) of two obliques meeting the plane at un- 
equal distances from the foot of the perpendicular, 
the more remote is the longer. 

If two lines are cut by three parallel planes, their cor- 
responding segments are proportional. 

Between two lines not in the same plane there is one 
common perpendicular, and only one. 

The bases of a cylinder are congruent. 

If a plane intersects a sphere, the line of intersection 
is a circle. 

The volumes of two tetrahedrons that have a trihedral 
angle of one equal to a trihedral angle of the other 
are to each other as the products of the three edges of 
these trihedral angles. 

In any polyhedron the number of edges increased by 
two is equal to the number of vertices increased by 
the number of faces. 

Two similar polyhedrons can be separated into the 
same number of tetrahedrons similar each to each and 
similarly placed. 


31 


41. The volumes of two similar tetrahedrons are to each 
other as the cubes of any two corresponding edges. 

42. The volumes of two similar polyhedrons are to each 
other as the cubes of any two corresponding edges. 

45. If three face angles of one trihedral angle are equal, 
respectively, to the three face angles of another the tri- 
hedral angles are either congruent or symmetric. 

44. Two spherical triangles on the same sphere are either 
congruent or symmetric if (a) two sides and the in- 
cluded angle of one are equal to the corresponding parts 
of the other; (b) two angles and the included side of one 
are equal to the corresponding parts of the other; (¢) 
they are mutually equilateral; (d) they are mutually 
equiangular. 

45. The sum of any two face angles of a trihedral angle is 

ereater than the third face angle. 

46, The sum of the face angles of any convex polyhedral 
angle is less than four right angles. 

47. Tach s‘de of a spherical triangle is less than the sum 
of the other two sides. 

48. The sum of the sides of a spherical polygon is less than 
360°. 

49, The sum of the angles of a spherical triangle is greater 
than 180° and less than 540°, 

50. There can not be more than five regular polyhedrons. 
D1. The locus of points equidistant (a) from two given 
points; (b) from two given planes which intersect. 
1V. SUBSIDIARY PROPOSITIONS IN MENSURATION 
52. The volume of a frustum of (a) a pyramid or (b) a 

cone. 

53. The lateral area of a frutsum of (a) a pyramid or (b) a 


cone of revolution. 
54. The volume of a prismo‘d (without formal proof). 


ADVANCED ALGEBRA 


1 Prerequisites. The work recommended under this caption will 
ordinarily best be given after the pupil has taken the work outlined 
for the first course in algebra, and a course in plane geometry. This 
will permit of a wide choice of interesting problem material and 
applications of algebra to the field of geometry. 


a Simple functions of one variable. Numerous illustrations and 
problems involving linear, quadratic and other simple functions in- 
~cCuding formulas from science and from common life. More diffi- 
cult problems in variation than those included in the earlier course. 


ye) 
0) a 
The following may serve to indicate the nature of material ap- 
propriate for inclusion here: 


Problems in variation such as those concerning wind pressures 
as dependent on wind-velocity, of fuel-consumption as dependent 
on velocity of travel, of strength of beams as dependent on dimen- 
sions of cross-sections afford opportunities for applying the equation 
Baan (. 

b Equation in one unknown. Various methods for solving a 
quadratic equation (such as factoring, completing the square, use 
of formula) should be given, but the method by completing the square 
sheuld be recognized as fundamental. 

In connection with the treatment of the quadratic a very brief 
ciscussion of complex numbers should be included. 

Treatment of radical equations should be restricted to the solu- 
tion of equations not more complicated than: 

pA ey ae lL ieee, 

By means of the graphic solution of equations of degree higher 
than the second, the pupil’s confidence in his ability to find an ap- 
proximate solution of any equation may be developed. Thus the 
problem “Find the size of the square to be cut from each corner of a 
sheet of tin 8” x 10”, in order to make an open box of volume 40 cu. 
in.” leads to a cubic equation whose approximate solution may be 
found from the graph. 


The solution of problems involving the idea of maximum and mini- 
mum may also be taken up in this connection. 


c Hquations in two or three unknowns. The algebraic solution 
of linear equations in two or three unknowns and the graphic solu- 
tion of linear equations in two unknowns should be given. The 
eraphie and algebraic solution of a linear and a quadratic equation 
and of two quadratics that contain no first degree term and no ry 
term should be included. 


d Haponents, radicals and logarithms. The definitions of nega- 
tive, zero and fractional exponents should be given and it should be 
made clear that these definitions must be adopted in order to make 
such exponents conform to the laws for positive integral exponents. 
Reduction of radical expressions to those involving fractional ex- 
ponents should be given as well as the inverse transformation. The 
fundamenta’ operations on expressions involving radicals, and such 
transformation as: 


a _a(Vb-Vo) 


n . af n BELBOS n n ey 
Vialb= 5. Mabey Maid Ob), aay oe 


‘popnput oq plnoys 


wre 


33 


In this connection, care should be taken to make clear the ad- 
vantage gained by such transformations. To “rationalize the deno- 
minator” without knowing why the operation is performed, is profit- 
less labor. In close connection with the work on exponents and , 
radicals there should be given as much of the theory of logarithms’ 
as is needed for their application to computation and sufficient prac- 
tice in their use in computation to impart a fair degree of facility. 


The graph of the equation y=10", with » taking value from 0 to 1, 
offers a good approach to the treatment of logarithms. Let n be 
successively 1, 4, 4, 4, 2, 3, 3, $, and compute the corresponding 
value of y, by taking the square root of ten, the square root of this 
result, et cetera. From the graph the pupil constructs, embodying 
these results, he can read the number corresponding to any value 
of n, and the value of n corresponding to any number, thus enabling 
him to perform multiplications and divisions by means of the 
graph. The printed tables should then be taken up. 


e Arithmetic and Geometric Progressions. The formulas for the 
nth term and the sum of n terms should be derived and applied to 
significant problems, such as computing the value of savings de- 
posited reguiarly, and the payment of debts in equal periodical sums, 
as in purchasing a home on monthly installments. 


f Binomial Theorem. A proof for positive integral exponents 
should be given and it should be stated that the formula holds for 
negative and fractional exponents under suitable restrictions, but 
proofs for these cases should be omitted. The problems may include 
the use of the formula for these exponents as well as positive in- 
tegral exnonents: as, for instanze, in the derivation of the approxi- 
mation formula: 


Vare+ b= (a2 +b)? =at =z 
TRIGONOMETRY 


Introductory work. The course in trigonometry presupposes that 
the pupil has had work such as is indicated in the section on numeri- 
cal trigonometry, on page 14. If such work has not already entered 
into the pupil’s mathematics course, it should be given as the intro- 
duction to the present course, with a time-allotment of about six 
weeks, 


In particular, the pupil should learn the definitions of sine, ‘cosine 
and tangent as pure numbers, depending only on the size of the 
angle. His first contact with these ratios will be made through 
measurement and tabulation from his own drawings of right tri- 
angles, The pupil should make his own table for a few angles by 


34 


graphical construction and computation to two places of decimals, 
before using the printed tables. He will learn to find any function 
when one is known, by drawing a right triangle, marking two sides 
to correspond to the numerator and denominator of the given fune- 
tion, computing the remaining side by the pythagorean theorem, and 
then reading off the new desired function directly from the figure. 


Problems and measurements. The solution of problems in in- 
direct measurement of heights and distances, based on measure- 
ments actually made by the pupils, will make them familiar with 
the complete solution of a right triangle by the aid of a table of 
sines, cosines and tangents when any two parts (including at least 
one side and excluding the right angle) are given. If possible, the 
transit should be used to carry out some of the simpler operations: 
of surveying. When no transit is available, apparatus should be: 
improvised and used for measuring angles. 


Diagrams. Every problem should be accompanied by a sketch or 
diagram to ensure that the pupil understands the meaning of each 
step of the work. An accurate graphical solution may be used as a 
check on the correctness of the numerical computation. 


Neatness. Neatness and systematic arrangement of work should 
be insisted upon from the beginning of the course. 


The course in trigonometry will be continued beyond the intro- 
ductory work above referred to, by considering the following topics: 


A Angles in general, and functions of any angle. Heretofore the 
angles considered have been largely acute angles, and the functions 
considered have been defined only for acute angles. It will be nec- 
essary to extend the idea of angle to cover the idea of angular mag- 
nitude in general, and to extend the idea of sine, cosine, and tangent 
to apply to angles other than those in the first quadrant. In this 
connection, it becomes necessary to fix clearly the algebraic signs of 
the various functions. For this purpose, the line representations of 
the functions in the unit-circle are probably most efficient. 


B Use of tables: reduction to the first quadrant. The tables 
vive values of the functions in the first quadrant only. It is neces- 
sary for the pupil now to become able to find the sine, cosine and 
tangent of angle in any quadrant: that is, to derive equations of the 
type cos (90 + A) = — sin A, and for this purpose, again, the line- 
representations in the unit circle are probably most useful. 


C Theory and use of logarithms. The work with logarithms may 
wisely minimize the theory. Four-place tables should be used. The 
slide rule is recommended for use in computation and checking, 


oy 


35 


D Trigonometric Hquations. The solution of trigonometrie 
equations should be restricted to those of about the order of diffi- 
culty of: 


» 
» 


sin x —2'cos x =2; and tan (x— 45) = ctn x 


Ee Formulas needed for the solution of problems. The pupil 
should be made to see that all oblique triangles can be solved by 
dissecting them into right triangles, and solving the latter in detail 
by applying the methods already learned. He should be led to see 
that the cumbersome and tedious solutions that are made necessary 
by this procedure can be replaced by more elegant and expeditious 
solutions when certain formulas, such as the “law of sines” and the 
“law of cosines,” are available. 

By leading the pupil to see the advantages to be gained by the use 
of logarithms in the solution of problems he will appreciate the 


A r 
value of such formulas as tan — = -—— 

2 s-a 
Throughout the work in trigonometry, stress should be laid upon 
problem-solving and work on formulas should be kept from degener- 
ating into purposeless juggling with symbols, by making clear the 
need for each group of formulas derived, and the advantage to be 
gained by their use. 

The teacher will easily find numerous applications of trigonom- 
etric formulas to engineering, navigation, and surveying. Problems 
secured from architects and engineering designers, involving appli- 
cations of trigonometry, are enormously more valuable for class use 
than mere text-book problems. Pupils should be encouraged to seek 
such problems for class use. 


FORMULAS 


A The following formulas should become part of the pupil's 
mental equipment, on a par with the multiplication tables: 


sin @ COS & 
I tan c—-———_ ; 1 o=-——_; 
sin @ 
"gee a= - ese o—=— 
COS @ sin @& 


Il The Pythagorean formulas: 


sin*® x -- cos? x = 1 
tan? x + 1 a= WeC* xX 
1 + ¢tn? x aa ee 


36 


Ill The Addition formulas: 
sin(#-+y)—sin & cos y+ cos & Sin y 
cos (#-+-y)—COS # COS y—Sin x Sin y 
tan #-+tan y 


tan(O+ V7 Fay fan y 


IV The Doubdle-angle formulas: 
sin 27=—=2-sin & cos & 
cos 2a—=cos’ #—s1n* & 
—2 cos? e—l 
—1—2 sin’ & 


V a The Law of Sines: 


Oa heey Gas tage 
snd sinB  sinQ 
bh The Law of Cosines: 
a?—b?-c*—2b¢e cos A. 


B The following sets of formulas are very useful, but not sufli- 
ciently important to justify the requirement that they be memo- 


rized: 


VI Formulas read off from a figure, as: 


cos (90°-+4a), sin (—a), et cetera, in terms of functions of x. 


VII The Half-Angle Formulas: 


sin y=2-sin - COs 4 
1-+-cos y=2 cos? sf 
1—cos y=2 sin? +; 

4] Se 
ere yj 2008 y 
» 2 

; ; ee 
néaee Ao 
9 2 
tan 4! con y 1 

9 peachees ¥ 


i, 


37 
VIII Special formulas for solving triangles: 


a Law of tangents: 
a—b tan %4(A—B) 


a+b tan %4(A+B) 
b When three sides are known: 


A | (s—b) (s—e) 
tan —=— | 
2 \ s(s—a) 


where s=!4(a-+6b-+-c) 


TWELFTH YEAR MATHEMATICS 


Opportunity to continue the study of mathematics beyond trig- 
onometry should be presented by providing courses such as the 
following: 


Elementary Statistics. Meaning and the use of fundamental 
concepts, simple frequency distributions with graphic representa- 
tions of various kinds, measure of central tendency. 


Mathematics of Finance. Interest, annuities, sinking funds, depre- 
ciation, the mathematical theory of insurance, the mathematics of 
building and loan associations. 


Shop Mathematics. Graphical representation, logarithmic «oi. 
putation, the use of tables of various kinds, the slide rule, empirical 
formulas. (This course may profitably replace the conventional 
eleventh year course for some groups of pupils.) 


Descriptive Geometry. Involving codperation between the draw- 
ing-room and mathematical teachers; stress laid at first upon prin- 


ciples, rather than on highly finished plates. 


General Mathematics. A course in composite mathematics, involy- 
ing the fundamental topics of college algebra, trigonometry, analytic 
geometry and calculus. A number of good texts are available. To be 
recommended for such schools as have a strong teacher and a good 
group of twelfth-year pupils. 


Elementary Calculus. The general notion of a derivative as a 
limit; application of derivatives to easy problems in rates and in 
maxima and minima; simple cases of inverse problems, as finding 
distances from velocities ; approximate methods of summation leading 
to the notion of integration; application to simple cases of motion, 
area, volume and pressure. 

Not all of these subjects can be offered in all schools, nor would 
it be desirable to do so. While all of the subjects listed here can be 
studied profitably by pupils who have done the work outlined through 
the eleventh year, the pupil’s vocational or later educational needs, 


38 


as well as the interests of the pupils who “like” mathematics, must 
be considered in determining the courses to be offered in any part- 
icular school. 


¢) 


BIBLIOGRAPHY 


High school mathematics teachers should be familiar with as many 
of the following publications as possible: 


1 


g 


6 


~ 


The Reorganization of the First Courses in Secondary 
School Mathematics: Secondary School Circular No. 5, 
Bureau of Education, Washington, D. C. 1920. 

Junior High School Mathematics: Secondary School 
Circular, No. 6, Bureau of Education, Washington, D. 
C. 1920. 

Cardinal Principles of Secondary Education, Bulletin 
1920, No. 1, Bureau of Education, Washington, D. C. 
The Problem of Mathematics in Secondary Education, 
Bulletin 1920, No. 1, Bureau of Education, Washing- 
ton, D. C. 

Scientific Method in the Reconstruction of Ninth Grade 
Mathematics, H. O. Rugg and J. R. Clark: University 
of Chicago Press, 1918. ) 

Psychology of High School Subjects: C. H. Judd, 
Ginn & Co., 1915. — 

The Teaching of Algebra (including Trigonometry) 
Nunn: Longmans, 1914. 

The Teaching of Geometry: D. E. Smith: Ginn, 1911. s 
The Teaching of Mathematics in Secondary Schools, c 
Schultze, Maemillan, 1916. 

Fundamental Concepts of Algebra and Geometry: 
J. W. Young, Macmillan, 1911. 

How We Think: John Dewey: D. C. Heath & Co., 
1910. 

Introduction to Mathematical Literature: G. A. Mil- 
ler. 

The Human Worth of Rigorous Thinking: Keyser: 
Columbia University Press, 1916. 

Mathematical Education: Carson: Ginn & Co., 1913. 
A Study of Mathematical Education: Benchara Bran- 
ford: Clarendon Press, 1908. 

First Year Algebra Scales: H. G. Hotz: Teachers 
College Contributions to Education, No. 190, 1918. 
Experimental Tests of Mathematical Ability and Their 
Prognostic Value, Agnes L. Rogers: Teachers College 4 
Contributions to Education, No, 89, 1918, 


Ww 


High school mathematics teachers should be familiar with the 


39 
JOURNALS 


following monthly publications: 


1 


|) 


Go 


Mathematics Teacher: devoted to the interests of 
teachers of mathematics in the junior and senior high 
school. 

The School Review: devoted to the interests of the high 


‘school in general. 


The Mathematical Monthly: devoted primarily to the 
interests of the college teacher of mathematics, and 
having much material of interest to high school teach- 
ers. 

School Science and Mathematics: a journal for science 
and mathematics teachers in the high school. 


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